I'll admit that this story is qutie interesting... But in the beginning, you should try separating lines to start new paragraphs so it seems less 'jumbled'. Plus, rather saying 'the girl' over and over, you could try putting more use to the pronouns.

Shiki mentions picking up things for Joshua... Remind me, how is that possible again? If this is in the original universe, Joshua should have zero interaction with Neku and the others due to 'breaking the rules'. Plus, why does Shiki want to buy for Joshua? Does she blame him for what happened? Going over little cracks like this in a story helps put it together better.

Concerning Fallen Angels... I'm going to point out that the Angels have higher power over the Composer, Fallen or not. One thing to keep in mind is that even a Fallen Angel is not something (or someone) the almighty Composer can easily sweep off the streets. Do not treat them like Noise; give them some more threats and what not.

In your last scene, you lack a lot of pronouns and you should separate lines a bit more. It's hard to follow sometimes and in once part, we have Joshua speaking, then the next is Lyra. It's a bit confusing because in stories, we want to separate more than one character's speech from the rest so we can get a better idea as to what's going on.

Other than that, I'd say focus on details a bit more, lower on dialogue. Not so much where they outweigh, but enough so it's a well balanced story rather a dialogue fanfic.

But I do believe this fanfic could have some potential, so I added it to alerts.